Convergents and irrationality measures of logarithms

Abstract

We prove new irrationality measures with restricted denominators of the form $\mathrm{d}_{\lfloor\nu m\rfloor}^s B^m$ (where $B, m \in\mathbb{N}, \nu > 0$, $s\in\{0,1\}$ and $\mathrm{d}_m=\textup{lcm}\{1,2, \ldots, m\}$) for values of the logarithm at certain rational numbers $r>0$. In particular, we show that such an irrationality measure of $\log(r)$ is arbitrarily close to 1 provided $r$ is sufficiently close to 1. This implies certain results on the number of non-zero digits in the $b$--ary expansion of $\log(r)$ and on the structure of the denominators of convergents of $\log(r)$. No simple method for calculating the latter is known. For example, we show that, given integers $a,c\ge 1$, for all large enough $b, n$, the denominator $q_n$ of the $n$--th convergent of $\log(1\pm a/b)$ cannot be written under the form $\dd_{\lfloor\nu m\rfloor}^s (bc)^m$: this is true for $a=c=1$, $b \ge 12$ when $s=0$, resp. $b \ge 2$ when $s=1$ and $\nu=1$. Our method rests on a detailed diophantine analysis of the upper Padé table $([p/q])_{p\ge q\ge 0}$ of the function $\log(1-x)$. Finally, we remark that worse results (of this form) are currently provable for the exponential function, despite the fact that the complete Padé table $([p/q])_{p, q\ge 0}$ of $\exp(x)$ and the convergents of $\exp(1/b)$, for $\vert b\vert \ge 1$, are well-known, for example.